Carleson-type removability for $p$-parabolic equations

TL;DR

This paper characterizes removable sets for H"older continuous solutions to degenerate p-parabolic equations using an intrinsic parabolic Hausdorff measure, introducing new methods applicable to a broad class of operators.

Contribution

It provides a necessary and sufficient condition for removability based on a novel measure, with a new proof technique relying on obstacle problems and supersolutions.

Findings

Characterization of removable sets via intrinsic parabolic Hausdorff measure.

New method for proving sufficiency based on obstacle problems.

Establishment of H"older continuity for solutions with measure data.

Abstract

We characterize removable sets for H\"older continuous solutions to degenerate parabolic equations of $p$-growth. A sufficient and necessary condition for a set to be removable is given in terms of an intrinsic parabolic Hausdorff measure, which depends on the considered H\"older exponent. We present a new method to prove the sufficient condition, which relies only on fundamental properties of the obstacle problem and supersolutions, and applies to a general class of operators. For the necessity of the condition, we establish the H\"older continuity of solutions with measure data, provided the measure satisfies a suitable decay property. The techniques developed in this article provide a new point of view even in the case $p=2$.